What are modern state-of-the-art solutions in prediction of time-series?
An interesting model I encountered in a course is Facebook Prophet. Prophet takes into account trends, seasonality, and holidays for its predictions. As you can probably guess, this is a model that fits Facebook's needs very well. I'll give a brief introduction then provide a link where you can read more. Prophet fits a couple of functions of time represented by a few terms. The general form of the timeseries predictions are, 

$$y(t)=g(t)+s(t)+h(t)+\epsilon_t$$

$g(t)$ deals with the trends I mentioned above. This is exactly what you would think it is, and accounts for non-periodic features of the data. It takes the form of a piecewise linear or logistic function.

$s(t)$ accounts for seasonality - in other words these are periodic changes in our timeseries (maybe an increase of sunscreen purchases in the summer). As this is periodic, the natural way to model this term is with Fourier decomposition to identify important frequencies in the signal.

$h(t)$ deals with predictable changes in the timeseries but is for events like holidays (this can happen at different times year to year so this is not necessarily periodic, think Easter). The user provides a list of events and how they want to account for it.

$\epsilon_t$ is just an error term to deal with anything that cannot be addressed with the rest of the model. 

This page has a great explanation if you want more detail. I highly recommend you check it out because it is very cool!